The Poisson-Lomax The Poisson-Lomax Distribution Distribuciأ³n Poisson-Lomax Bander Al-Zahrania, Hanaa

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  • Revista Colombiana de Estadística Junio 2014, volumen 37, no. 1, pp. 223 a 243

    The Poisson-Lomax Distribution

    Distribución Poisson-Lomax

    Bander Al-Zahrania, Hanaa Sagorb

    Department of Statistics, King Abdulaziz University, Jeddah, Saudi Arabia

    Abstract

    In this paper we propose a new three-parameter lifetime distribution with upside-down bathtub shaped failure rate. The distribution is a com- pound distribution of the zero-truncated Poisson and the Lomax distribu- tions (PLD). The density function, shape of the hazard rate function, a general expansion for moments, the density of the rth order statistic, and the mean and median deviations of the PLD are derived and studied in de- tail. The maximum likelihood estimators of the unknown parameters are obtained. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance-covariance matrix. Finally, a real data set is analyzed to show the potential of the new proposed distribution.

    Key words: Asymptotic variance-covariance matrix, Compounding, Life- time distributions, Lomax distribution, Poisson distribution, Maximum like- lihood estimation.

    Resumen

    En este artículo se propone una nueva distribución de sobrevida de tres parámetros con tasa fallo en forma de bañera. La distribución es una mezcla de la Poisson truncada y la distribución Lomax. La función de densidad, la función de riesgo, una expansión general de los momentos, la densidad del r-ésimo estadístico de orden, y la media así como su desviación estándar son derivadas y estudiadas en detalle. Los estimadores de máximo verosímiles de los parámetros desconocidos son obtenidos. Los intervalos de confianza asintóticas se obtienen según la matriz de varianzas y covarianzas asintótica. Finalmente, un conjunto de datos reales es analizado para construir el po- tencial de la nueva distribución propuesta.

    Palabras clave: mezclas, distribuciones de sobrevida, distribució n Lomax, distribución Poisson, estomación máximo-verosímil.

    aProfessor. E-mail: bmalzahrani@kau.edu.sa bPh.D student. E-mail: hsagor123@gmail.com

    223

  • 224 Bander Al-Zahrani & Hanaa Sagor

    1. Introduction

    Marshall & Olkin (1997) introduced an effective technique to add a new pa- rameter to a family of distributions. A great deal of papers have appeared in the literature used this technique to propose new distributions. In their paper, Marshall & Olkin (1997) generalized the exponential and Weibull distributions. Alice & Jose (2003) followed the same approach and introduced Marshall-Olkin extended semi-Pareto model and studied its geometric extreme stability. Ghitany, Al-Hussaini & Al-Jarallah (2005) studied the Marshall-Olkin Weibull distribution and established its properties in the presence of censored data. Marshall-Olkin extended Lomax distribution was introduced by Ghitany, Al-Awadhi & Alkhalfan (2007). Compounding Poisson and exponential distributions have been considered by many authors; e.g. Kus (2007) proposed the Poisson-exponential lifetime distri- bution with a decreasing failure rate function. Al-Awadhi & Ghitany (2001) used the Lomax distribution as a mixing distribution for the Poisson parameter and ob- tained the discrete Poisson-Lomax distribution. Cancho, Louzada-Neto & Barriga (2011) introduced another modification of the Poisson-exponential distribution.

    Let Y1, Y2, . . . , YZ be independent and identically distributed random variables each has a density function f , and let Z be a discrete random variable having a zero-truncated Poisson distribution with probability mass function

    PZ(z) ≡ PZ(z, λ) = e−λλz

    z!(1− e−λ) , z ∈ {1, 2, . . .}, λ > 0. (1)

    Suppose that X is a random variable representing the lifetime of a parallel-system of Z components, i.e. X = max{Y1, Y2, . . . , Yz}, and Y ’s and Z are independent. The conditional distribution function of X|Z has the probability density function (pdf)

    fX|Z(x|z) = zf(x)[F (x)]z−1. (2)

    where F (x) is the cumulative distribution function (cdf) corresponding to f(x).

    A compound probability function (pdf) of fX|Z(x|z) and PZ(z), where X is a continuous random variable (r.v.) and Z a discrete r.v. is defined by

    gX(x) =

    ∞∑ z=1

    fX|Z(x|z)PZ(z). (3)

    Substitution of (1) and (2) in (3) then yields

    gX(x) =

    ∞∑ z=1

    zf(x)[F (x)]z−1 (

    λze−λ

    z!(1− e−λ)

    ) =

    λf(x)e−λ(1−F (x))

    (1− e−λ) , x > 0, λ > 0.

    Revista Colombiana de Estadística 37 (2014) 223–243

  • The Poisson-Lomax Distribution 225

    The reliability and the hazard rate functions of X are, respectively, given by

    Ḡ(x, λ) = 1− e−λF̄ (x)

    (1− e−λ) , x > 0, (4)

    hG(x, λ) = λf(x)e−λF̄ (x)

    1− e−λF̄ (x) =

    λf(x)

    eλF̄ (x) − 1 . (5)

    In this paper we propose a new lifetime distribution by compounding Poisson and Lomax distributions. As we have mentioned in the previous chapters, the Lomax distribution with two parameters is a special case of the generalized Pareto distribution, and ti is also known as the Pareto of the second type. A random variable X is said to have the Lomax distribution, abbreviated as X ∼ LD(α, β), if it has the pdf

    fLD(x;α, β) = αβ (1 + βx) −(α+1)

    , x > 0, α, β > 0. (6)

    Here α and β are the shape and scale parameters, respectively. Analogous tu above, the survival and hazard functions associated with (6) are given by

    F̄LD(x;α, β) = (1 + βx) −α

    , x > 0, (7)

    hLD(x;α, β) = αβ

    1 + βx , x > 0. (8)

    The rest of the paper is organized as follows. In Section 2, we give explicit forms and interpretation for the distribution function and the probability density func- tion. In Section 3, we discuss the distributional properties of the proposed dis- tribution. Section 4 discusses the estimation problem using the maximum likeli- hood estimation method. In Section 5, an illustrative example, model selections, goodness-of-fit tests for the distribution with estimated parameters are all pre- sented. Finally, we conclude in Section 6.

    2. Model Formulation

    Substitution of (7) in (4) yields the following reliability function:

    Ḡ(x;α, β, λ) = 1− e−λ(1+βx)−α

    (1− e−λ) , x > 0, α, β, λ > 0. (9)

    The pdf associated with (9) is expressed in a closed form and is given by

    g(x;α, β, λ) = αβλ (1 + βx)

    −(α+1) e−λ(1+βx)

    −α

    (1− e−λ) , x > 0, α, β, λ > 0. (10)

    The density function given by (10) can be interpreted as a compound of the zero- truncated Poisson distribution and the Lomax distribution. Suppose that X = max{Y1, Y2, · · · , Yz}, and each Y is distributed according to the Lomax distribtion.

    Revista Colombiana de Estadística 37 (2014) 223–243

  • 226 Bander Al-Zahrani & Hanaa Sagor

    The variable Z has zero-truncated Poisson distribution and the variables Y ’s and Z are independent. Then the conditional distribution function of X|Z has the pdf

    fX|Z(x|z;α, β) = zαβ(1 + βx)−(α+1)[1− (1 + βx)−α]z−1. (11)

    The joint distribution of the random variables X and Z, denoted by fX,Z(x, z), is given by

    fX,Z(x, z) = z

    z!(1− e−λ) αβ(1 + βx)−(α+1)[1− (1 + βx)−α]z−1e−λλz, (12)

    the marginal pdf of X is as follows.

    fX(x;α, β, λ) = αβλe−λ(1 + βx)−(α+1)

    (1− e−λ)

    ∞∑ z=1

    [(1− (1 + βx)−α)λ]z−1

    (z − 1)!

    = αβλe−λ(1 + βx)−(α+1)eλ(1−(1+βx)

    −α)

    (1− e−λ)

    = αβλ(1 + βx)−(α+1)e−λ(1+βx)

    −α

    (1− e−λ) ,

    which is the distribution with the pdf given by (10). The distribution of X may be referred to as the Poisson-Lomax distribution. Symbolically it is abbreviated by X ∼ PLD(α, β, λ) to indicate that the random variable X has the Poisson-Lomax distribution with parameters α, β and λ.

    3. Distributional Properties

    In this section, we study the distributional properties of the PLD. In particular, if X ∼ PLD(α, β, λ) then the shapes of the density function, the shapes of the hazard function, moments, the density of the rth order statistics, and the mean and median deviations of the PLD are derived and studied in detail.

    3.1. Shapes of pdf

    The limit of the Poisson-Lomax density as x→∞ is 0 and the limit as x→ 0 is αβλ/(eλ − 1). The following theorem gives simple conditions under which the pdf is decreasing or unimodal.

    Theorem 1. The pdf, g(x), of X ∼ PLD(α, β, λ) is decreasing (unimodal) if the function ξ(x) ≥ 0 (< 0) where ξ(x) = α(1− λ(1 + βx)−α) + 1, independent of β.

    Proof . The first derivative of g(x) is given by

    g′(x) = − αβ 2λ

    1− e−λ (1 + βx)−(α+2) e−λ(1+βx)

    −α ξ((1 + βx)−α),

    where ξ(y) = α(1−λy) + 1, and y = (1 +βx)−α < 1. Then we have the following:

    Revista Colombiana de Estadística 37 (2014) 223–243

  • The Poisson-Lomax Distribution 227

    (i) If ξ(1) = α(1− λ) + 1 > 0, then ξ(y) > 0 for all y < 1, and hence, g′(x) ≤ 0 for all x > 0, i.e. the function g(x) is decreasing.

    (ii) If ξ(1) < 0, then ξ(y) has a unique zero at yξ = α+1αλ < 1 . Since y = (1 + βx)−α is one to one transformation, it follows that g(x) has also a unique critical point at xg